A more thorough understanding has developed through the lens of optimisation. Already in [1]} it was shown that each iteration of AdaBoost decreases the training error. Moreover, in [2]}, [3]}, a close connection to the exponential loss was pointed out and studied. Building on these results, [4]}, [5]}, [6]} discovered that overparameterized AdaBoost, when run long enough with vanishing learning rate \(\epsilon \) , has \(\ell _1\) -margin converging to the max-\(\ell _1\) -margin. In particular, this means that given training data \((Y_i, X_i)_{i=1}^n\) , where the \(Y_i\)  are binary and the \(X_i\)  are \(p\) -dimensional feature vectors, and where \(\hat{\beta }_T\)  denotes the output of AdaBoost with the canonical basis as simple classifiers, learning rate \(\epsilon \)  and run-time \(T\) , we have
\( \min _{1 \le i \le n} \frac{Y_i \langle X_i, \hat{\beta }_T\rangle }{\Vert \hat{\beta }_T\Vert _1} \xrightarrow{} \max _{\beta \ne 0} \min _{1 \le i \le n} \frac{Y_i \langle X_i, \beta \rangle }{\Vert \beta \Vert _1}=:\gamma ,\) 
